Introduction
The purpose of this document is to provide a reproducible record of all analyses and figures in the main article. The main article is focused on the effect of response diversity on community stability in fluctuating environments. We are going to look at the effect of response diversity, richness, temperature and nutrients on community temporal stability. Specifically, we are going to look at the effect of fundamental balance (our measurement of stability) on temporal stability. Then, as response diversity is thought to stabilize temporal stability of aggregate community properties via asynchrony, we are going to look at the relationship between response diversity and asynchrony.
Finally, as multiple evidence suggests that compensatory dynamics and temporal stability are determine by species interactions, we are going to analyse the effect of species interactions on stability to understand if they are more important than response diversity in driving temporal stability of community biomass.
This document is produced by an Rmarkdown file that includes code to reproduce from data all results presented in the main article.
Load datasets
Data wrangling and balance calculation
# needs to have id_new variable
ciliate_traits <- ciliate_traits %>%
dplyr::mutate(
# Remove dots from the date
cleaned_date = gsub("\\.", "", date),
# Extract the part of id after the underscore
id_suffix = sub(".*_(.*)", "\\1", id),
# Combine cleaned_date, id_suffix, and species_initial into a new variable
id_new = paste0(cleaned_date, id_suffix, composition)
) %>%
# Optionally, remove the intermediate columns to clean up
dplyr::select(-cleaned_date, -id_suffix,-new_id)
uniqueN(ciliate_traits$id_new)==nrow(ciliate_traits) # all unique ;)
id_dd<-full_join(dd_all_pred,dplyr::select(ciliate_traits,id_new,biomass),join_by("id_new"))
## add day variable
#create a day variable from the date variable
id_dd<-dplyr::mutate(id_dd,date=as.Date(date,format = "%d.%m.%y"))
earliest_date<-min(id_dd$date)
days_since_earliest<-as.numeric(id_dd$date-earliest_date)+1
id_dd<-id_dd%>%dplyr::mutate(day=days_since_earliest)
#create a summarised df on microcosm level with each species seperate
# Make sure, that we have n_frames and not N_frames
names(id_dd)[names(id_dd) == "N_frames"] <- "n_frames"
#extrapolation_factor <- 9.301902 # for 16 x magnification
extrapolation_factor <- 9.828125 # for 25 x magnification
video_biomass_species <- c( "C", "P", "S","D","L","T")
biomasses <- id_dd %>%
dplyr::group_by( day,temperature,nutrients,sample_ID,composition,predict_spec) %>% # group by xxx
dplyr::summarize(
biomass = sum(biomass * n_frames, na.rm = TRUE) / (1 * 125) # if not 3 videos corrections is done below with dens_factor
) %>%
dplyr::mutate(
biomass = biomass * extrapolation_factor,
)
biomasses<-biomasses%>%dplyr::mutate(biomass=biomass*1000)
dd_ts_id<-biomasses
#fill up missing dates with biomass<-0
fill_dd<-expand.grid(sample_ID=unique(dd_ts_id$sample_ID),day=unique(dd_ts_id$day),predict_spec=unique(dd_ts_id$predict_spec))
complete_ts<-full_join(fill_dd,dd_ts_id,join_by(sample_ID,day,predict_spec))
complete_ts$biomass[is.na(complete_ts$biomass)]<-0
complete_ts<-complete_ts%>%dplyr::mutate(composition=sub("_.*", "", sample_ID))
complete_ts<-complete_ts %>%
dplyr::mutate(temperature = sapply(strsplit(as.character(sample_ID), "_"), function(x) paste(x[3], x[4], sep = "-")))
complete_ts<- dplyr::mutate(complete_ts,nutrients = gsub(".*Nut(.*?)_.*", "\\1", sample_ID))
# Now remove wrong combinations of composition and predict_spec / predict_spec
complete_ts<- complete_ts %>%
rowwise() %>%
dplyr::filter(predict_spec %in% unlist(strsplit(composition, ""))) %>%
ungroup()
complete_ts<-dplyr::mutate(complete_ts,temperature=as.character(temperature),
nutrients=as.character(nutrients),
richness=nchar(composition))
complete_ts<-complete_ts%>%group_by(sample_ID,composition,day)%>%dplyr::mutate(tot_biomass=sum(biomass))
complete_ts<-complete_ts%>%dplyr::mutate(biom_contribution=biomass/tot_biomass)
df_biomass_mod <- complete_ts
complete_ts<-complete_ts%>%dplyr::mutate(temperature=paste0(temperature," °C"),
nutrients=paste0(nutrients," g/L"))
# introduce slopes of
names(df_slopes)[names(df_slopes)=="species_initial"]<-"predict_spec"
slope_ts<-full_join(dplyr::select(df_slopes,nutrients,predict_spec,temperature,slope),complete_ts)
slope_ts<-slope_ts%>%dplyr::mutate(w_slope=biom_contribution*slope,
sign=sign(slope))
slope_ts<-slope_ts%>%group_by(sample_ID,temperature,nutrients,richness,composition,day,tot_biomass)%>%dplyr::summarize(
sum_w_slopes=abs(sum(w_slope)),
mean_abs_slope=mean(abs(slope)),
sum_abs_slope=sum(abs(slope)),
abs_sum_slope=abs(sum(slope)),
symmetry=abs(sum(sign)))
slope_ts<-slope_ts%>%dplyr::mutate(richness=as.factor(richness))
##create new variable where it checks, where the last observation =0 is; with complete_ts
aggr_ts <- slope_ts %>%
group_by( sample_ID) %>%
arrange(day) %>%
mutate(
# Create a flag for non-zero tot_biomass
non_zero_biomass = tot_biomass != 0,
# Find the last non-zero day
last_non_zero_day = ifelse(any(non_zero_biomass), max(day[non_zero_biomass], na.rm = TRUE), NA),
# Find the first zero day after the last non-zero day
first_zero_day = ifelse(
!is.na(last_non_zero_day),
min(day[!non_zero_biomass & day > last_non_zero_day], na.rm = TRUE),
NA
),
# Flag for days after the first zero day
is_after_first_zero_day = ifelse(!is.na(first_zero_day), day > first_zero_day, FALSE)
) %>%
ungroup()
aggr_ts<-aggr_ts%>%mutate(rep_var=sub("_[^_]+$", "", sample_ID))
biomass_ts<-aggr_ts%>%group_by(day,temperature,nutrients,richness)%>%summarize(tot_biom=mean(tot_biomass),se_tot_biom=sd(tot_biomass)/sqrt(as.numeric(length(tot_biomass))))
Biomass
Let’s have a look at the biomass dynamics in the different environmental treatments.
tot biomass plot
Figure 1 : Community total biomass during the experiment in different environmental treatments. Different color represent richness levels.
Main Results
We now look at the main results of the experiment. We are going to look first at the effect of richness, temperature and nutrients on community temporal stability. Then, we are going to look at the relationship between divergence (original response diversity metric) and temporal stability. Finally, we are going to look at the relationship between response diversity and temporal stability.
In the whole analysis, we calculated the temporal stability of total community biomass as the inverse of the coefficient of variation (ICV) (i.e. 1/\(\frac{\sigma}{\mu}\)).
Effect of T, N and R
Figure 2: Effects of richness (a), temperature (b), and nutrients (c) on community total biomass temporal stability.
We can see that richness does not have a clear effect on community temporal stability, while stability was higher at lower temperature, and nutrients increased community temporal stability.
Effect of Divergence
We look at the relationship between divergence (our original response diversity metric) and stability
Figure 3: Relationship between Divergence and temporal stability of total community biomass.
Divergence is positively related to temporal stability, suggesting that response diversity promotes stability. However, the relationship between divergence and stability becomes weaker as richness increases. We think that this is due to divergence considering only the responses of the 2 most “responding” species. Thus, when species richness increases, disregarding the responses of the other species in the community except the 2 responding the most makes the relationship between response diversity and stability weaker.
This is why, after running the experiment, we developed another metric to measure response diversity, which we called balance, and that is presented in the main text of the publication.
Balance has several desirable features that makes it a more suitable metric than divergence: Independence of richness, higher predictive power, and accounts for the responses of all species in the community (as opposed to divergence that accounts for only the 2 most “responding” species).
Here, we provide extensive evidence of why balance is a better metric to measure response diversity than divergence, and thus justifying focusing the analysis around balance.
Comparing Divergence and Balance
We first compare how well divergence and balance predict stability (predictive power).
Balance
#
mod1 <- lm(data=complete_aggr,log10(stability)~log10(balance_f))
Divergence
mod2 <- lm(data=complete_aggr,log10(stability)~(divergence))
Table 1: Comparision of model performance of divergence and balance as predictors of stability. Model 1 has balance as predictor and model 2 has divergence as predictor.
|
model
|
AIC
|
AICc
|
BIC
|
R2
|
R2_adjusted
|
RMSE
|
Sigma
|
|
1
|
-89.27328
|
-89.17286
|
-78.79409
|
0.1917679
|
0.1884142
|
0.1510344
|
0.1516599
|
|
2
|
-55.71579
|
-55.61538
|
-45.23661
|
0.0720796
|
0.0682293
|
0.1618316
|
0.1625017
|
A model with Balance as predictor performs better than one with divergence as predictor, and it explains more of the variance in stability than divergence.
Moreover, from Figure 3, it looks like divergence declines in performance as richness increases. Let’s test this analytically.
To do than we build a linear model having stability as response variable and either log10(balance) or divergence as predictor for each richness level. We then extract the R squared of the models and their standardised estimates. (standardised estimates were calculated centering divergence and balance using the function scale()).
# getting model estimates for each richness level
lm_divergence_richness_E <- complete_aggr %>%
nest(data = -richness) %>%
mutate(
model = map(data, ~ lm(log10(stability) ~ scale(divergence), data = .x)),
results = map(model, broom::tidy)
) %>%
unnest(results) %>% dplyr::filter(term=="scale(divergence)")
# getting model R squared for each richness level
lm_divergence_richness_R <- complete_aggr %>%
nest(data = -richness) %>%
mutate(
model = map(data, ~ lm(log10(stability) ~ scale(divergence), data = .x)),
results = map(model, broom::glance)
) %>%
unnest(results)
lm_divergence_richness_R
## # A tibble: 3 × 15
## richness data model r.squared adj.r.squared sigma statistic p.value df
## <fct> <list> <list> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2 <tibble> <lm> 0.201 0.191 0.209 19.9 2.66e-5 1
## 2 3 <tibble> <lm> 0.172 0.161 0.108 16.4 1.19e-4 1
## 3 4 <tibble> <lm> 0.0337 0.0215 0.129 2.76 1.01e-1 1
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## # df.residual <int>, nobs <int>
# getting model estimatesf or each richness level
lm_balance_richness_E <- complete_aggr %>%
nest(data = -richness) %>%
mutate(
model = map(data, ~ lm(log10(stability) ~ scale(log10(balance_f)), data = .x)),
results = map(model, broom::tidy)
) %>%
unnest(results) %>% dplyr::filter(term=="scale(log10(balance_f))")
# getting model R squared for each richness level
lm_balance_richness_R <- complete_aggr %>%
nest(data = -richness) %>%
mutate(
model = map(data, ~ lm(log10(stability) ~ scale(log10(balance_f)), data = .x)),
results = map(model, broom::glance)
) %>%
unnest(results)
lm_balance_richness_R
## # A tibble: 3 × 15
## richness data model r.squared adj.r.squared sigma statistic p.value df
## <fct> <list> <list> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 2 <tibble> <lm> 0.188 0.177 0.210 18.2 5.36e-5 1
## 2 3 <tibble> <lm> 0.232 0.222 0.104 23.8 5.44e-6 1
## 3 4 <tibble> <lm> 0.272 0.263 0.112 29.6 5.84e-7 1
## # ℹ 6 more variables: logLik <dbl>, AIC <dbl>, BIC <dbl>, deviance <dbl>,
## # df.residual <int>, nobs <int>
Figure 4: Performance comparison of divergence vs balance. In (a), the R squared of linear models for divergence and balance are shown for each richness level. In (b), the estimates of the linear models for divergence and balance are shown for each richness level.
We can see that the R squared of divergence as predictor of stability becomes smaller as richness increases, while the R squared of balance as predictor of stability does not (actually increases slightly).
Now we build a linear model were stability is modeled as a function of balance and divergence.
Then, we compared the variance explained by the full model compared to a model containing either only balance or only divergence.
lm_div_balance <- lm(data=complete_aggr,log10(stability)~log10(balance_f)+divergence)
model with only divergence
lm_div <- lm(data=complete_aggr,log10(stability)~divergence)
model with only balance
lm_balance <- lm(data=complete_aggr,log10(stability)~log10(balance_f))
Table 2: Comparison of model performance of divergence, balance and both as predictors of stability. Model 1 has both balance and divergence as predictors, model 2 has divergence as predictor, and model 3 has balance as predictor.
|
model
|
AIC
|
AICc
|
BIC
|
R2
|
R2_adjusted
|
RMSE
|
Sigma
|
|
1
|
-88.97683
|
-88.80876
|
-75.00458
|
0.1974141
|
0.1907259
|
0.1505060
|
0.1514437
|
|
1
|
-55.71579
|
-55.61538
|
-45.23661
|
0.0720796
|
0.0682293
|
0.1618316
|
0.1625017
|
|
2
|
-89.27328
|
-89.17286
|
-78.79409
|
0.1917679
|
0.1884142
|
0.1510344
|
0.1516599
|
Type III anova table: a model with both balance and divergence as predictors is not significantly different from a model with only balance as predictor.
anova(lm_div_balance, lm_balance)
## Analysis of Variance Table
##
## Model 1: log10(stability) ~ log10(balance_f) + divergence
## Model 2: log10(stability) ~ log10(balance_f)
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 240 5.5044
## 2 241 5.5432 -1 -0.038724 1.6884 0.1951
Type III anova table: a model with both balance and divergence as predictors is significantly better from a model with only divergence as predictor.
anova(lm_div_balance, lm_div)
## Analysis of Variance Table
##
## Model 1: log10(stability) ~ log10(balance_f) + divergence
## Model 2: log10(stability) ~ divergence
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 240 5.5044
## 2 241 6.3640 -1 -0.85959 37.479 3.741e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Overall, balance explains more of the variance in stability than divergence, and there is virtually no difference between a model containing only balance and the full model.
Interaction divergence and richness
Richness had to transformed to numeric and to be centered to avoid collinearity with divergence
lm_rich_div <- lm(data=complete_aggr,log10(stability)~divergence*scale(as.numeric(richness)))
Type III anova table
anova(lm_rich_div)
## Analysis of Variance Table
##
## Response: log10(stability)
## Df Sum Sq Mean Sq F value Pr(>F)
## divergence 1 0.4943 0.49435 19.8282 1.301e-05 ***
## scale(as.numeric(richness)) 1 0.1558 0.15579 6.2487 0.013100 *
## divergence:scale(as.numeric(richness)) 1 0.2496 0.24958 10.0106 0.001758 **
## Residuals 239 5.9587 0.02493
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Divergence significantly interact with richness, suggesting that the relationship between divergence and stability changes with richness.
While an ideal metric of response diversity should be independent of richness.
We repeat the same model using balance instead of divergence.
lm_rich_balance <- lm(data=complete_aggr,log10(stability)~log10(balance_f)*scale(as.numeric(richness)))
Type III anova table
anova(lm_rich_balance)
## Analysis of Variance Table
##
## Response: log10(stability)
## Df Sum Sq Mean Sq F value
## log10(balance_f) 1 1.3152 1.31522 57.1274
## scale(as.numeric(richness)) 1 0.0003 0.00028 0.0122
## log10(balance_f):scale(as.numeric(richness)) 1 0.0405 0.04050 1.7589
## Residuals 239 5.5024 0.02302
## Pr(>F)
## log10(balance_f) 8.694e-13 ***
## scale(as.numeric(richness)) 0.9123
## log10(balance_f):scale(as.numeric(richness)) 0.1860
## Residuals
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Balance does not significantly interact with richness, suggesting that the relationship between balance and stability is stable across richness levels.
Finally, we assess variable importance using the relative importance of predictors in the full model.
We use the package vip (https://cran.r-project.org/web/packages/vip/vignettes/vip.html) to calculate the relative importance of predictors in the full model.
The function vip::vip for multiple linear regression, or linear models (LMs), uses the absolute value of the -statistic as a measure of VI.
Motivation for the use of the associated 𝑡-statistic is given in Bring (1994) [https://www.tandfonline.com/doi/abs/10.1080/00031305.1994.10476059].
vip::vip(lm_div_balance)
Figure 5: Variable importance in the model including both balance and divergence as predictors of stability.
We believe that the extensive evidence here provided justifies focusing the analysis around balance, and not divergence, as a metric of response diversity.
We will thus only look at balance for the rest of the analysis.
Effect RD
We are now going to look at how response diversity (balance) affected temporal stability of total community biomass. We are going to look at the relationship between fundamental balance (so based only on species response surfaces measured in monoculture), an realised balance (measured accounting for species contribution to balance).
This is fundamentally testing our most important hypothesis.
Figure 6: Effects of fundamental and realised response diversity (measured as balance) on total community biomass temporal stability.
We can see that balance is always negatively related to temporal stability, which means that response diversity promotes stability across richness levels. Interestingly, we see that there is little difference between fundamental and realised balance. Yet, as the richness increases, the relationship between realised balance and stability becomes steeper compared to fundamental balance.
Linear models
Model: Fundamental balance
First we analyze the effect of fundamental balance, temperature, nutrients and richness on biomass temporal stability using a linear model.
balance was modelled as continuous variables, while richness, temperature and nutrients were modelled as categorical variables. balance and stability were log-transformed to meet the assumptions of linear models.
lm_full<-lm(data=complete_aggr,log10(stability)~log10(balance_f)+(richness)+nutrients+temperature)
summary(lm_full)
##
## Call:
## lm(formula = log10(stability) ~ log10(balance_f) + (richness) +
## nutrients + temperature, data = complete_aggr)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.29981 -0.07245 -0.00896 0.04696 0.41550
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.34893 0.02822 -12.367 < 2e-16 ***
## log10(balance_f) -0.05352 0.01596 -3.354 0.000929 ***
## richness3 -0.04166 0.01849 -2.253 0.025193 *
## richness4 -0.01244 0.01871 -0.665 0.506925
## nutrients0.35 g/L 0.17958 0.01860 9.656 < 2e-16 ***
## nutrients0.75 g/L 0.21218 0.01932 10.982 < 2e-16 ***
## temperature22-25 °C -0.07781 0.01853 -4.199 3.81e-05 ***
## temperature25-28 °C -0.09830 0.02457 -4.002 8.44e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1177 on 235 degrees of freedom
## Multiple R-squared: 0.5256, Adjusted R-squared: 0.5115
## F-statistic: 37.2 on 7 and 235 DF, p-value: < 2.2e-16
anova(lm_full)
## Analysis of Variance Table
##
## Response: log10(stability)
## Df Sum Sq Mean Sq F value Pr(>F)
## log10(balance_f) 1 1.3152 1.31522 94.9984 < 2.2e-16 ***
## richness 2 0.0850 0.04251 3.0706 0.04826 *
## nutrients 2 1.8765 0.93823 67.7684 < 2.2e-16 ***
## temperature 2 0.3282 0.16410 11.8530 1.246e-05 ***
## Residuals 235 3.2535 0.01384
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
A linear model was fitted to examine the effects of resource balance, richness, nutrients, and temperature on community stability (measured as log₁₀(stability)). The model explained a significant portion of the variance (Adjusted R² = 0.5115, F(7, 235) = 37.2, p < 2.2e-16).
The intercept of the model was estimated at -0.349 (SE = 0.028, p < 2e-16), indicating the baseline log₁₀(stability) when all predictor variables are at their reference levels.
Among the predictors, log₁₀(balance) showed a significant negative effect on stability (Estimate = -0.054, SE = 0.016, p = 0.0009). This suggests that as balance increases (more balance), stability tends to decrease.
In terms of richness, communities with three species (richness3) showed a significant decrease in stability compared to the reference level (richness2; 𝛽=−0.042±0.018β=−0.042±0.018; t=−2.25, p=0.025). In contrast, communities with four species (richness4) did not differ significantly from the reference level (𝛽=−0.012±0.019β=−0.012±0.019; t=−0.67, p=0.51).
In contrast, communities with four species (richness4) did not differ significantly from the reference level (𝛽=−0.012±0.019β=−0.012±0.019; t=−0.67, p=0.51).
Nutrient concentration also had a significant positive effect on stability, with estimates for 0.35 g/L (Estimate = 0.180, SE = 0.019, p < 2e-16) and 0.75 g/L (Estimate = 0.212, SE = 0.019, p < 2e-16) indicating increased stability with higher nutrient levels.
Finally, temperature regimes showed a significant effect on stability. Both 22–25 °C (Estimate = -0.078, SE = 0.019, p = 3.81e-05) and 25–28 °C (Estimate = -0.098, SE = 0.025, p = 8.44e-05) significantly reduced stability when compared to the baseline (18–21 °C).
In summary, our findings show that temporal stability is significantly influenced by response diversity (balance), nutrient concentration, and temperature, with higher nutrient concentrations enhancing stability and higher temperatures reducing it. However, species richness was not a significant determinant of stability within the conditions of this study.
Prepare publication-ready table
Summary table
Table 2: Linear model results for the effects of balance, richness, nutrients, and temperature on community stability. Estimates are presented with 95% confidence intervals and p-values. Significant results are highlighted in bold.
|
Term
|
Estimate
|
Lower 95% CI
|
Upper 95% CI
|
t value
|
p-value
|
|
(Intercept)
|
-0.349
|
-0.405
|
-0.293
|
-12.367
|
2.17e-27
|
|
log10(balance_f)
|
-0.054
|
-0.085
|
-0.022
|
-3.354
|
9.29e-04
|
|
richness3
|
-0.042
|
-0.078
|
-0.005
|
-2.253
|
2.52e-02
|
|
richness4
|
-0.012
|
-0.049
|
0.024
|
-0.665
|
5.07e-01
|
|
nutrients0.35 g/L
|
0.180
|
0.143
|
0.216
|
9.656
|
8.60e-19
|
|
nutrients0.75 g/L
|
0.212
|
0.174
|
0.250
|
10.982
|
6.43e-23
|
|
temperature22-25 °C
|
-0.078
|
-0.114
|
-0.041
|
-4.199
|
3.81e-05
|
|
temperature25-28 °C
|
-0.098
|
-0.147
|
-0.050
|
-4.002
|
8.44e-05
|
Table 3:Pairwise contrasts
| Predictor |
Linear Regression Results
|
95% CI |
Linear Regression Results
|
| Estimate |
p-value |
| log10(balance_f) |
-0.05 |
-0.08, -0.02 |
<0.001 |
| richness |
|
|
|
| richness3 - richness2 |
-0.04 |
-0.09, 0.00 |
0.065 |
| richness4 - richness2 |
-0.01 |
-0.06, 0.03 |
0.8 |
| richness4 - richness3 |
0.03 |
-0.01, 0.07 |
0.3 |
| nutrients |
|
|
|
| (0.35 g/L) - (0.01 g/L) |
0.18 |
0.14, 0.22 |
<0.001 |
| (0.75 g/L) - (0.01 g/L) |
0.21 |
0.17, 0.26 |
<0.001 |
| (0.75 g/L) - (0.35 g/L) |
0.03 |
-0.01, 0.08 |
0.2 |
| temperature |
|
|
|
| (22-25 °C) - (18-21 °C) |
-0.08 |
-0.12, -0.03 |
<0.001 |
| (25-28 °C) - (18-21 °C) |
-0.10 |
-0.16, -0.04 |
<0.001 |
| (25-28 °C) - (22-25 °C) |
-0.02 |
-0.08, 0.04 |
0.7 |
sum vs. weighted_sum
We now look at how fundamental and realised balance are related to each other.
Figure 7: Relationship between fundamental and realised balance. The black line represents the 1:1 relationship between fundamental and realised balance.
Asynchrony
Response diversity (aka balance) has been suggested as a mechanism that promotes temporal stability of community biomass by promoting species asynchrony.
We thus calculated the asynchrony index suggested by Gross et al. (2014)[https://www.journals.uchicago.edu/doi/epdf/10.1086/673915] to calculate the effect of asynchrony on temporal stability and to see how reponse diversity relate to asynchrony.
The index ranges between -1 and 1, with -1 indicating perfect asyncrony and 1 being perfectly synchronous, and 0 indicating random variation.
Plot stability vs. Asynchrony Gross
Figure 8: Relationship between temporal stability and asynchrony (Gross) divided by nutrient level.
Plot Asynchrony Gross vs fundamental balance
Figure 9: Relationship between asynchrony (Gross) and fundamental balance divided by nutrient level.
SEM
Now, we use a structural equation model (SEM) to explore how stability is influenced by asynchrony, temperature, nutrient levels, balance, and richness, with asynchrony also modeled as dependent on balance, nutrients, and richness.
model1B <- '
stability ~ asynchrony_Gross
+temperature
+nutrients
+log_balance_f+
richness
asynchrony_Gross ~ log_balance_f
+nutrients
+richness
'
# Fit the model
fit1B <- sem(model1B, estimator="MLM",meanstructure = TRUE,data = sem_aggr%>%dplyr::filter(!is.na(synchrony_Gross)))
# Summarize the results
summary(fit1B, standardized = TRUE,rsquare=T, fit.measures = TRUE)
## lavaan 0.6-19 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 12
##
## Number of observations 241
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 1.777 1.537
## Degrees of freedom 1 1
## P-value (Chi-square) 0.183 0.215
## Scaling correction factor 1.156
## Satorra-Bentler correction
##
## Model Test Baseline Model:
##
## Test statistic 295.357 353.081
## Degrees of freedom 9 9
## P-value 0.000 0.000
## Scaling correction factor 0.837
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 0.997 0.998
## Tucker-Lewis Index (TLI) 0.976 0.986
##
## Robust Comparative Fit Index (CFI) 0.998
## Robust Tucker-Lewis Index (TLI) 0.981
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) 152.706 152.706
## Loglikelihood unrestricted model (H1) NA NA
##
## Akaike (AIC) -281.411 -281.411
## Bayesian (BIC) -239.594 -239.594
## Sample-size adjusted Bayesian (SABIC) -277.631 -277.631
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.057 0.047
## 90 Percent confidence interval - lower 0.000 0.000
## 90 Percent confidence interval - upper 0.192 0.176
## P-value H_0: RMSEA <= 0.050 0.306 0.356
## P-value H_0: RMSEA >= 0.080 0.531 0.469
##
## Robust RMSEA 0.051
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.200
## P-value H_0: Robust RMSEA <= 0.050 0.327
## P-value H_0: Robust RMSEA >= 0.080 0.525
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.012 0.012
##
## Parameter Estimates:
##
## Standard errors Robust.sem
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## stability ~
## asynchrny_Grss 0.195 0.033 5.947 0.000 0.195 0.399
## temperature -0.049 0.011 -4.454 0.000 -0.049 -0.241
## nutrients 0.147 0.011 13.216 0.000 0.147 0.719
## log_balance_f -0.029 0.012 -2.452 0.014 -0.029 -0.120
## richness 0.011 0.010 1.181 0.238 0.011 0.055
## asynchrony_Gross ~
## log_balance_f -0.080 0.029 -2.719 0.007 -0.080 -0.159
## nutrients -0.210 0.022 -9.373 0.000 -0.210 -0.501
## richness -0.102 0.023 -4.404 0.000 -0.102 -0.243
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .stability -0.407 0.040 -10.122 0.000 -0.407 -2.441
## .asynchrny_Grss 0.125 0.065 1.921 0.055 0.125 0.367
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .stability 0.012 0.001 10.670 0.000 0.012 0.430
## .asynchrny_Grss 0.081 0.009 8.917 0.000 0.081 0.694
##
## R-Square:
## Estimate
## stability 0.570
## asynchrny_Grss 0.306
Model Fit
The model fit indices suggest a good fit:
Comparative Fit Index (CFI) = 0.998 and Tucker-Lewis Index (TLI) = 0.986, both indicating a good fit as values close to 1 are considered strong.
Root Mean Square Error of Approximation (RMSEA) = 0.047 (with robust RMSEA at 0.051) and Standardized Root Mean Square Residual (SRMR) = 0.012. These values indicate a good fit, with RMSEA and SRMR values below 0.05 generally preferred.
Interpretation of Pathways
Stability:
Asynchrony: Positive and highly significant effect on stability, suggesting that asynchrony (indicating lack of synchrony or compensatory dynamics) is associated with greater stability.
Temperature: Negative effect, where higher temperature values correlate with lower stability, potentially due to physiological stress or disruption in community dynamics at higher temperatures.
Nutrients: Positive and highly significant, suggesting that greater nutrient availability enhances stability, possibly through support for higher productivity or resource buffering.
balance: Negative and significant effect, where greater balance reduces stability.
Richness: Not significant, indicating that within this model, richness does not have a notable effect on stability.
Asynchrony:
balance: Negative and significant, suggesting that greater balance reduces asynchrony.
Nutrients: Negative and highly significant effect, indicating that higher nutrient concentrations are associated with lower asynchrony, possibly due to homogenizing effects of nutrient availability.
Richness: Negative and significant, where increased richness is associated with reduced asynchrony, possibly indicating increased interactions or overlap in resource use among species.
Explained Variance
Stability: The model explains 57% of the variance in stability, suggesting a substantial amount of stability is accounted for by these factors.
Asynchrony: The model explains 30.6% of the variance in asynchrony, indicating that while balance, nutrients, and richness contribute, other factors may also play a role in driving asynchrony.
Summary
This SEM model demonstrates that stability in the ecosystem is positively associated with asynchrony and nutrient levels, but negatively associated with temperature and balance. Interestingly, species richness has no direct impact on stability but does reduce asynchrony, indicating indirect complexity in the stability-dynamics relationship. This highlights the role of environmental and community factors in ecosystem stability, with asynchrony serving as a crucial intermediary in maintaining stability in fluctuating conditions.
Figure 10: Structural equation model (SEM) of the relationship between fundamental balance, asynchrony, richness, nutrients, temperature and temporal stability. The model shows that fundamental balance has a negative effect on asynchrony, which in turn has a positive effect on temporal stability. The model also shows that fundamental balance has a direct negative effect on temporal stability. Temperature has a direct negative effect on temporal stability, while nutrients have a direct positive effect on temporal stability. Richness has a direct negative effect on asynchrony, but no direct effect on temporal stability.
Species Interactions
lm_M_int<-lm(data=int_aggr%>%dplyr::filter(theta%in%c("none","var")),log10(stability)~log10(balance_f)+nutrients+temperature+(richness)+mean_interaction)
summary(lm_M_int)
##
## Call:
## lm(formula = log10(stability) ~ log10(balance_f) + nutrients +
## temperature + (richness) + mean_interaction, data = int_aggr %>%
## dplyr::filter(theta %in% c("none", "var")))
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.29920 -0.07316 -0.00719 0.05333 0.41059
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.355922 0.028377 -12.543 < 2e-16 ***
## log10(balance_f) -0.054555 0.015899 -3.431 0.000710 ***
## nutrients0.35 g/L 0.192131 0.019863 9.673 < 2e-16 ***
## nutrients0.75 g/L 0.222239 0.020080 11.068 < 2e-16 ***
## temperature22-25 °C -0.072852 0.018669 -3.902 0.000125 ***
## temperature25-28 °C -0.092956 0.024651 -3.771 0.000206 ***
## richness3 -0.039305 0.018459 -2.129 0.034277 *
## richness4 -0.003528 0.019315 -0.183 0.855244
## mean_interaction -0.049311 0.028230 -1.747 0.081987 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1172 on 234 degrees of freedom
## Multiple R-squared: 0.5317, Adjusted R-squared: 0.5157
## F-statistic: 33.21 on 8 and 234 DF, p-value: < 2.2e-16
anova(lm_M_int)
## Analysis of Variance Table
##
## Response: log10(stability)
## Df Sum Sq Mean Sq F value Pr(>F)
## log10(balance_f) 1 1.3152 1.31522 95.8276 < 2.2e-16 ***
## nutrients 2 1.8824 0.94120 68.5761 < 2.2e-16 ***
## temperature 2 0.3333 0.16663 12.1409 9.623e-06 ***
## richness 2 0.0740 0.03701 2.6968 0.06952 .
## mean_interaction 1 0.0419 0.04188 3.0512 0.08199 .
## Residuals 234 3.2116 0.01372
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# create new variable describing the level of balance_f. Categories should be low, medium_low, medium_high, high and based on 25% quantiles
int_aggr<-int_aggr%>%mutate(balance_level=ifelse(balance_f<quantile(balance_f,0.25),"low",
ifelse(balance_f<quantile(balance_f,0.5),"medium_low",
ifelse(balance_f<quantile(balance_f,0.75),"medium_high","high"))))
# create variable low_balance that is 1 if balance_level is low or medium_low and 0 otherwise
int_aggr%>%dplyr::filter(theta%in%c("none","var"))%>%dplyr::mutate(low_balance=ifelse(balance_level%in%c("low","medium_low"),1,0))%>%
dplyr::mutate(neg_syn=ifelse(synchrony_Gross<0,1,0))%>%
dplyr::summarize(perc_neg_syn=sum(neg_syn)/n(),perc_neg_int=sum(low_balance)/n())
## # A tibble: 1 × 2
## perc_neg_syn perc_neg_int
## <dbl> <dbl>
## 1 NA 0.506
# calculate percentage of negative mean_interaction associated with negative synchrony_gross
int_aggr%>%dplyr::filter(theta%in%c("none","var"))%>%dplyr::mutate(neg_int=ifelse(mean_interaction<0,1,0))%>%
dplyr::mutate(neg_syn=ifelse(synchrony_Gross<0,1,0))%>%
dplyr::summarize(perc_neg_syn=sum(neg_syn)/n(),perc_neg_int=sum(neg_int)/n())
## # A tibble: 1 × 2
## perc_neg_syn perc_neg_int
## <dbl> <dbl>
## 1 NA 0.0700
Figure 11: Relationship between mean interaction strength and asynchrony (Gross) divided by nutrient level.
Figure 12: Relationship between mean interaction strength and stability divided by nutrient level.
Figure 13: Relationship between mean interaction strength and asynchrony (Gross) divided by nutrient level.
Figure 14: Distribution of mean interaction coefficient faceted by temperature, nutrients, and richness.
SEM2
sem_aggr2 <- int_aggr %>%
ungroup() %>% # Ensure there is no grouping
mutate(
log_balance_f = log10(balance_f),
stability = log10(1 / CV),
richness = as.numeric(richness),
temperature=temperature,
asynchrony_Gross= (-synchrony_Gross)
#Keep it as an ordered factor
)
model1B <- '
stability ~ asynchrony_Gross
+nutrients
+log_balance_f+
richness +
mean_interaction+
temperature
asynchrony_Gross ~ log_balance_f
+nutrients+
richness+
mean_interaction
'
# Fit the model
fit1B <- sem(model1B, estimator="MLM",meanstructure = TRUE,data = sem_aggr2%>%dplyr::filter(!is.na(asynchrony_Gross), theta%in%c("none","var")))
# Summarize the results
summary(fit1B, standardized = TRUE,rsquare=T, fit.measures = TRUE)
## lavaan 0.6-19 ended normally after 1 iteration
##
## Estimator ML
## Optimization method NLMINB
## Number of model parameters 14
##
## Number of observations 241
##
## Model Test User Model:
## Standard Scaled
## Test Statistic 0.661 0.534
## Degrees of freedom 1 1
## P-value (Chi-square) 0.416 0.465
## Scaling correction factor 1.238
## Satorra-Bentler correction
##
## Model Test Baseline Model:
##
## Test statistic 315.059 427.408
## Degrees of freedom 11 11
## P-value 0.000 0.000
## Scaling correction factor 0.737
##
## User Model versus Baseline Model:
##
## Comparative Fit Index (CFI) 1.000 1.000
## Tucker-Lewis Index (TLI) 1.012 1.012
##
## Robust Comparative Fit Index (CFI) 1.000
## Robust Tucker-Lewis Index (TLI) 1.021
##
## Loglikelihood and Information Criteria:
##
## Loglikelihood user model (H0) 163.115 163.115
## Loglikelihood unrestricted model (H1) NA NA
##
## Akaike (AIC) -298.229 -298.229
## Bayesian (BIC) -249.442 -249.442
## Sample-size adjusted Bayesian (SABIC) -293.819 -293.819
##
## Root Mean Square Error of Approximation:
##
## RMSEA 0.000 0.000
## 90 Percent confidence interval - lower 0.000 0.000
## 90 Percent confidence interval - upper 0.158 0.137
## P-value H_0: RMSEA <= 0.050 0.541 0.632
## P-value H_0: RMSEA >= 0.080 0.314 0.228
##
## Robust RMSEA 0.000
## 90 Percent confidence interval - lower 0.000
## 90 Percent confidence interval - upper 0.170
## P-value H_0: Robust RMSEA <= 0.050 0.563
## P-value H_0: Robust RMSEA >= 0.080 0.318
##
## Standardized Root Mean Square Residual:
##
## SRMR 0.006 0.006
##
## Parameter Estimates:
##
## Standard errors Robust.sem
## Information Expected
## Information saturated (h1) model Structured
##
## Regressions:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## stability ~
## asynchrny_Grss 0.206 0.034 6.096 0.000 0.206 0.421
## nutrients 0.145 0.011 13.381 0.000 0.145 0.707
## log_balance_f -0.027 0.012 -2.355 0.019 -0.027 -0.112
## richness 0.009 0.010 0.892 0.373 0.009 0.042
## mean_interactn 0.043 0.022 1.907 0.056 0.043 0.077
## temperature -0.051 0.011 -4.655 0.000 -0.051 -0.250
## asynchrony_Gross ~
## log_balance_f -0.075 0.028 -2.701 0.007 -0.075 -0.150
## nutrients -0.180 0.024 -7.441 0.000 -0.180 -0.430
## richness -0.078 0.021 -3.675 0.000 -0.078 -0.186
## mean_interactn -0.272 0.054 -5.034 0.000 -0.272 -0.242
##
## Intercepts:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .stability -0.394 0.041 -9.713 0.000 -0.394 -2.361
## .asynchrny_Grss 0.071 0.066 1.082 0.279 0.071 0.208
##
## Variances:
## Estimate Std.Err z-value P(>|z|) Std.lv Std.all
## .stability 0.012 0.001 10.476 0.000 0.012 0.424
## .asynchrny_Grss 0.075 0.008 8.996 0.000 0.075 0.644
##
## R-Square:
## Estimate
## stability 0.576
## asynchrny_Grss 0.356
Figure 15: Structural equation model (SEM) of the relationship between fundamental balance, asynchrony, richness, species interactions, nutrients, temperature and temporal stability.
Model Fit
Chi-Square Test: The model test statistic for the user model (𝜒2=0.661,df=1, p=0.416) suggests that the model fits the data well, as the p-value is not significant. This indicates no significant difference between the observed and predicted covariance matrices.
Comparative Fit Index (CFI) and Tucker-Lewis Index (TLI): Both indices are very close to 1 (CFI = 1.000, TLI = 1.012), indicating excellent model fit.
RMSEA: The RMSEA is 0, with a 90% confidence interval from 0 to 0.158. The low RMSEA, along with p-values for tests of close fit (RMSEA ≤ 0.05), further supports a well-fitting model.
SRMR: The SRMR value of 0.006 indicates a good fit, as values below 0.08 are generally acceptable.
Regression Paths
Predictors of Stability
Asynchrony_Gross: Positively associated with stability (β=0.206, p<0.001, standardized = 0.421), suggesting that greater asynchrony among species contributes to higher stability.
Nutrients: Significant positive association with stability (β=0.145,p<0.001, standardized = 0.707), indicating that increased nutrient levels promote stability.
Log(Balance): A small but significant negative association (β=−0.027, p=0.019, standardized = -0.112), suggesting that greater balance in response diversity slightly decreases stability.
Richness: Not a significant predictor (β=0.009, p=0.373).
Mean Interaction: Approaches significance (β=0.043, p=0.056), suggesting a potential weak positive influence on stability.
Temperature: Shows a significant negative association (β=−0.051, p<0.001, standardized = -0.250), indicating that higher temperatures tend to reduce stability.
Predictors of Asynchrony_Gross
Log(Balance): Shows a negative relationship with asynchrony (β=−0.075, p=0.007), suggesting that higher balance is associated with lower asynchrony among species.
Nutrients: Significant negative effect (β=−0.180, p<0.001, standardized = -0.430), indicating that higher nutrient levels reduce species asynchrony.
Richness: Negatively associated with asynchrony (β=−0.078, p<0.001), implying that greater species richness leads to less asynchrony.
Mean Interaction: A strong negative association with asynchrony (β=−0.272, p<0.001), suggesting that higher mean interaction values reduce species asynchrony.
Variance and R-Squared
Overall, the model suggests that stability is positively influenced by asynchrony and nutrient levels, while it decreases with higher temperatures and increased balance in species responses. Asynchrony itself is reduced by increased nutrients, richness, balance, and mean interaction levels, highlighting complex interactions between community structure and environmental factors that influence ecosystem stability.
